arXiv subjects

Francesco Bullo

Publications and source records attributed to Francesco Bullo.

At least 109 records · Page 6Linked to original sources

arXiv2020

Graph-Theoretic Stability Conditions for Metzler Matrices and Monotone Systems

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we derive two sets of stability conditions based on two forms of input-to-state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. Based on the max-interconnection gains, we show that the cyclic small-gain theorem becomes necessary and sufficient for the stability of Metzler systems; based on the sum-interconnection gains, we obtain novel graph-theoretic conditions for the stability of Metzler systems. All these conditions highlight the role of cycles in the interconnection graph and unveil how the structural properties of the graph affect stability. Finally, we extend our results to the nonlinear monotone system and obtain similar sufficient conditions for global asymptotic stability.

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arXiv2020

Singular Perturbation and Small-signal Stability for Inverter Networks

This paper examines small-signal stability of electrical networks composed dominantly of three-phase grid-following inverters. We show that the mere existence of a high-voltage power flow solution does not necessarily imply small-signal stability; this motivates us to develop a framework for stability analysis that systematically acknowledges inverter dynamics. We identify a suitable time-scale decomposition for the inverter dynamics, and using singular perturbation theory, obtain an analytic sufficient condition to verify small-signal stability. Compared to the alternative of performing an eigenvalue analysis of the full-order network dynamics, our analytic sufficient condition reduces computational complexity and yields insights on the role of network topology and constitution as well as inverter-filter and control parameters on small-signal stability. Numerical simulations for a radial network validate the approach and illustrate the efficiency of our analytic conditions for designing and monitoring grid-tied inverter networks.

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arXiv2020

The 1995-2018 Global Evolution of the Network of Amicable and Hostile Relations Among Nation-States

We draw on the data collected by the Integrated Crisis Early Warning System on millions of international and regional public news stories, and this system's indicators of the orientation toward a specific nation-state. We construct the networks of international amicable and hostile relations among nation-states that occur in specific time-periods in order to study the global evolution of the network of such international appraisals. Our analysis presents evidence of an evolution of the structure of this network and a model of the probabilistic micro-dynamics of the alterations of international appraisals during the 1995-2018 span of the available data. Our research provides empirical findings on long-standing debates in the interdisciplinary field of work on Structural Balance Theory. Also remarkably, we find that the trajectory of the Frobenius norm of sequential transition probabilities, which govern the evolution of international appraisals among nations, dramatically stabilizes.

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arXiv2020

LaSalle Invariance Principle for Discrete-time Dynamical Systems: A Concise and Self-contained Tutorial

LaSalle invariance principle was originally proposed in the 1950's and has become a fundamental mathematical tool in the area of dynamical systems and control. In both theoretical research and engineering practice, discrete-time dynamical systems have been at least as extensively studied as continuous-time systems. For example, model predictive control is typically studied in discrete-time via Lyapunov methods. However, there is a peculiar absence in the standard literature of standard treatments of Lyapunov functions and LaSalle invariance principle for discrete-time nonlinear systems. Most of the textbooks on nonlinear dynamical systems focus only on continuous-time systems. In Chapter 1 of the book by LaSalle [11], the author establishes the LaSalle invariance principle for difference equation systems. However, all the useful lemmas in [11] are given in the form of exercises with no proof provided. In this document, we provide the proofs of all the lemmas proposed in [11] that are needed to derive the main theorem on the LaSalle invariance principle for discrete-time dynamical systems. We organize all the materials in a self-contained manner. We first introduce some basic concepts and definitions in Section 1, such as dynamical systems, invariant sets, and limit sets. In Section 2 we present and prove some useful lemmas on the properties of invariant sets and limit sets. Finally, we establish the original LaSalle invariance principle for discrete-time dynamical systems and a simple extension in Section~3. In Section 4, we provide some references on extensions of LaSalle invariance principles for further reading. This document is intended for educational and tutorial purposes and contains lemmas that might be useful as a reference for researchers.

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arXiv2020

Matching-Based Capture Strategies for 3D Heterogeneous Multiplayer Reach-Avoid Differential Games

This paper studies a 3D multiplayer reach-avoid differential game with a goal region and a play region. Multiple pursuers defend the goal region by consecutively capturing multiple evaders in the play region. The players have heterogeneous moving speeds and the pursuers have heterogeneous capture radii. Since this game is hard to analyze directly, we decompose the whole game as many subgames which involve multiple pursuers and only one evader. Then, these subgames are used as a building block for the pursuer-evader matching. First, for multiple pursuers and one evader, we introduce an evasion space (ES) method characterized by a potential function to construct a guaranteed pursuer winning strategy. Then, based on this strategy, we develop conditions to determine whether a pursuit team can guard the goal region against one evader. It is shown that in 3D, if a pursuit team is able to defend the goal region against an evader, then at most three pursuers in the team are necessarily needed. We also compute the value function of the Hamilton-Jacobi-Isaacs (HJI) equation for a special subgame of degree. To capture the maximum number of evaders in the open-loop sense, we formulate a maximum bipartite matching problem with conflict graph (MBMC). We show that the MBMC is NP-hard and design a polynomial-time constant-factor approximation algorithm to solve it. Finally, we propose a receding horizon strategy for the pursuit team where in each horizon an MBMC is solved and the strategies of the pursuers are given. We also extend our results to the case of a bounded convex play region where the evaders escape through an exit. Two numerical examples are provided to demonstrate the obtained results.

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arXiv2020

Stable and Efficient Structures in Multigroup Network Formation

In this work we present a strategic network formation model predicting the emergence of multigroup structures. Individuals decide to form or remove links based on the benefits and costs those connections carry; we focus on bilateral consent for link formation. An exogenous system specifies the frequency of coordination issues arising among the groups. We are interested in structures that arise to resolve coordination issues and, specifically, structures in which groups are linked through bridging, redundant, and co-membership interconnections. We characterize the conditions under which certain structures are stable and study their efficiency as well as the convergence of formation dynamics.

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arXiv2020

Convergence Properties of the Heterogeneous Deffuant-Weisbuch Model

The Deffuant-Weisbuch (DW) model is a bounded-confidence opinion dynamics model that has attracted much recent interest. Despite its simplicity and appeal, the DW model has proved technically hard to analyze and its most basic convergence properties, easy to observe numerically, are only conjectures. This paper solves the convergence problem for the heterogeneous DW model with the weighting factor not less than $1/2$. We establish that, for any positive confidence bounds and initial values, the opinion of each agent will converge to a limit value almost surely, and the convergence rate is exponential in mean square. Moreover, we show that the limiting opinions of any two agents either are the same or have a distance larger than the confidence bounds of the two agents. Finally, we provide some sufficient conditions for the heterogeneous DW model to reach consensus.

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arXiv2019

Robotic Surveillance Based on the Meeting Time of Random Walks

This paper analyzes the meeting time between a pair of pursuer and evader performing random walks on digraphs. The existing bounds on the meeting time usually work only for certain classes of walks and cannot be used to formulate optimization problems and design robotic strategies. First, by analyzing multiple random walks on a common graph as a single random walk on the Kronecker product graph, we provide the first closed-form expression for the expected meeting time in terms of the transition matrices of the moving agents. This novel expression leads to necessary and sufficient conditions for the meeting time to be finite and to insightful graph-theoretic interpretations. Second, based on the closed-form expression, we setup and study the minimization problem for the expected capture time for a pursuer/evader pair. We report theoretical and numerical results on basic case studies to show the effectiveness of the design.

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arXiv2019

Polarization and Fluctuations in Signed Social Networks

Much recent research on social networks has focused on the modeling and analysis of how opinions evolve as a function of interpersonal relationships. It is also of great interest to model and understand the implications of friendly and antagonistic relationships. In this paper, we propose a new, simple and intuitive model that incorporates the socio-psychological phenomenon of the boomerang effect in opinion dynamics. We establish that, under certain conditions on the structure of the signed network that corresponds to the so-called structural balance property, the opinions in the network polarize. Compared to other models in the literature, our model displays a richer and perhaps more intuitive behavior of the opinions when the social network does not satisfy structural balance. In particular, we analyze signed networks in which the opinions show persistent fluctuations (including the case of the so-called clustering balance).

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arXiv2019

Signed Network Formation Games and Clustering Balance

We propose a signed network formation game, in which pairs of individuals strategically change the signs of the edges in a complete network. These individuals are members of a social network who strategically reduce cognitive dissonances by changing their interpersonal appraisals. We characterize the best-response dynamics for this game and prove that its implementation \pc{can} dynamically drive the network to a sociologically meaningful sign configuration called clustering balance. In this configuration, agents in the social network form one or more clusters that have positive relationships among their members but negative relationships among members of other clusters. In the past, various researchers in the fields of psycho-sociology, political science, and physics have looked at models that explain the generation of up to two clusters. Our work contributes to these fields by proposing a simple model that generates a broader class of signed networks.

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arXiv2019

Structural Balance via Gradient Flows over Signed Graphs

Structural balance is a classic property of signed graphs satisfying Heider's seminal axioms. Mathematical sociologists have studied balance theory since its inception in the 1040s. Recent research has focused on the development of dynamic models explaining the emergence of structural balance. In this paper, we introduce a novel class of parsimonious dynamic models for structural balance based on an interpersonal influence process. Our proposed models are gradient flows of an energy function, called the dissonance function, which captures the cognitive dissonance arising from violations of Heider's axioms. Thus, we build a new connection with the literature on energy landscape minimization. This gradient flow characterization allows us to study the transient and asymptotic behaviors of our model. We provide mathematical and numerical results describing the critical points of the dissonance function.

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arXiv2019

Dynamic Social Balance and Convergent Appraisals via Homophily and Influence Mechanisms

Social balance theory describes allowable and forbidden configurations of the topologies of signed directed social appraisal networks. In this paper, we propose two discrete-time dynamical systems that explain how an appraisal network \textcolor{blue}{converges to} social balance from an initially unbalanced configuration. These two models are based on two different socio-psychological mechanisms respectively: the homophily mechanism and the influence mechanism. Our main theoretical contribution is a comprehensive analysis for both models in three steps. First, we establish the well-posedness and bounded evolution of the interpersonal appraisals. Second, we fully characterize the set of equilibrium points; for both models, each equilibrium network is composed by an arbitrary number of complete subgraphs satisfying structural balance. Third, we establish the equivalence among three distinct properties: non-vanishing appraisals, convergence to all-to-all appraisal networks, and finite-time achievement of social balance. In addition to theoretical analysis, Monte Carlo validations illustrates how the non-vanishing appraisal condition holds for generic initial conditions in both models. Moreover, numerical comparison between the two models indicate that the homophily-based model might be a more universal explanation for the formation of social balance. Finally, adopting the homophily-based model, we present numerical results on the mediation and globalization of local conflicts, the competition for allies, and the asymptotic formation of a single versus two factions.

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arXiv2019

Finite-time influence systems and the Wisdom of Crowd effect

Recent contributions have studied how an influence system may affect the wisdom of crowd phenomenon. In the so-called naive learning setting, a crowd of individuals holds opinions that are statistically independent estimates of an unknown parameter; the crowd is wise when the average opinion converges to the true parameter in the limit of infinitely many individuals. Unfortunately, even starting from wise initial opinions, a crowd subject to certain influence systems may lose its wisdom. It is of great interest to characterize when an influence system preserves the crowd wisdom effect. In this paper we introduce and characterize numerous wisdom preservation properties of the basic French-DeGroot influence system model. Instead of requiring complete convergence to consensus as in the previous naive learning model by Golub and Jackson, we study finite-time executions of the French-DeGroot influence process and establish in this novel context the notion of prominent families (as a group of individuals with outsize influence). Surprisingly, finite-time wisdom preservation of the influence system is strictly distinct from its infinite-time version. We provide a comprehensive treatment of various finite-time wisdom preservation notions, counterexamples to meaningful conjectures, and a complete characterization of equal-neighbor influence systems.

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arXiv2019

Multi-group connectivity structures and their implications

We investigate the implications of different forms of multi-group connectivity. Four multi-group connectivity modalities are considered: co-memberships, edge bundles, bridges, and liaison hierarchies. We propose generative models to generate these four modalities. Our models are variants of planted partition or stochastic block models conditioned under certain topological constraints. We report findings of a comparative analysis in which we evaluate these structures, controlling for their edge densities and sizes, on mean rates of information propagation, convergence times to consensus, and steady state deviations from the consensus value in the presence of noise as network size increases.

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arXiv2019

UAV Surveillance Under Visibility and Dwell-Time Constraints: A Sampling-Based Approach

A framework is introduced for planning unmanned aerial vehicle flight paths for visual surveillance of ground targets, each having particular viewing requirements. Specifically, each target is associated with a set of imaging parameters, including a desired (i) tilt angle, (ii) azimuth, with the option of a 360-degree view, and (iii) dwell-time. Tours are sought to image the targets, while minimizing both the total mission time and the time required to reach the initial target. An epsilon-constraint scalarization is used to pose the multi-objective problem as a constrained optimization, which, through careful discretization, can be approximated as a discrete graph-search. It is shown that, in many cases, this approximation is equivalent to a generalized traveling salesperson problem. A heuristic procedure for solving the discrete approximation and recovering solutions to the full routing problem is presented, and is shown to have resolution completeness properties. Algorithms are illustrated through numerical studies.

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arXiv2019

Social power evolution in influence networks with stubborn individuals

This paper studies the evolution of social power in influence networks with stubborn individuals. Based on the Friedkin-Johnsen opinion dynamics and the reflected appraisal mechanism, two models are proposed over issue sequences and over a single issue, respectively. These models generalize the original DeGroot-Friedkin (DF) model by including stubbornness. To the best of our knowledge, this paper is the first attempt to investigate the social power evolution of stubborn individuals basing on the reflected appraisal mechanism. Properties of equilibria and convergence are provided. We show that the models have same equilibrium social power and convergence property, where the equilibrium social power depends only upon interpersonal influence and individuals' stubbornness. Roughly speaking, more stubborn individual has more equilibrium social power. Moreover, unlike the DF model without stubbornness, we prove that for the models with stubbornness, autocracy can never be achieved, while democracy can be achieved under any network topology.

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arXiv2018

Novel Multi Agent Models for Chemical Self-assembly

The chemical self-assembly has been considered as one of most important scientific problems in the 21th Century; however, since the process of self-assembly is very complex, there is few mathematic theory for it currently. This paper provides a novel multi-agent model for chemical self-assembly, where the interaction between agents adopts the classic Lennard-Jones potential. Under this model, we propose an optimal problem by taking the temperature as the control input, and choosing the internal energy as the optimal object. A numerical solution for our optimal problem is also developed. Simulations show that our control scheme can improve the product of self-assembly. Further more, we give a strict analysis for the self-assembly model without noise, which corresponds to an attraction-repulsion multi-agent system, and prove it converges to a stable configuration eventually.

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arXiv2018

Synchronization of Kuramoto Oscillators: Inverse Taylor Expansions

Synchronization in networks of coupled oscillators is a widely studied topic with extensive scientific and engineering applications. In this paper, we study the frequency synchronization problem for networks of Kuramoto oscillators with arbitrary topology and heterogeneous edge weights. We propose a novel equivalent transcription for the equilibrium synchronization equation. Using this transcription, we develop a power series expansion to compute the synchronized solution of the Kuramoto model as well as a sufficient condition for the strong convergence of this series expansion. Truncating the power series provides (i) an efficient approximation scheme for computing the synchronized solution, and (ii) a simple-to-check, statistically-correct hierarchy of increasingly accurate synchronization tests. This hierarchy of tests provides a theoretical foundation for and generalizes the best-known approximate synchronization test in the literature. Our numerical experiments illustrate the accuracy and the computational efficiency of the truncated series approximation compared to existing iterative methods and existing synchronization tests.

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